On Minimum Dominating Sets in cubic and (claw,H)-free graphs

02/27/2020

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by   Valentin Bouquet, et al.

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Given a graph G=(V,E), S⊆ V is a dominating set if every v∈ V∖ S is adjacent to an element of S. The Minimum Dominating Set problem asks for a dominating set with minimum cardinality. It is well known that its decision version is NP-complete even when G is a claw-free graph. We give a complexity dichotomy for the Minimum Dominating Set problem for the class of (claw, H)-free graphs when H has at most six vertices. In an intermediate step we show that the Minimum Dominating Set problem is NP-complete for cubic graphs.